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Questions tagged [bivariate-distributions]

For questions on bivariate distribution, the joint probability distribution of two random variables.

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Geometric interpretation of joint normal conditional expectation

Suppose $X,Y\sim N(\mu,\Sigma)$ are bivariate normal variables. For simplicity I'll assume $X$ and $Y$ are centered on the origin. I'm looking for a visual or geometric way to understand the quantity $...
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Finding the joint probability of a transformed standard normal

$(X, Y)$ follows a standard bivariate normal distribution with $-1<\rho<1$. Let $A = X + Y$ and $B = XY$. Find the conditional probability of $P(A > 0|B>0)$. Since $A$ follows standard ...
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How to show that $X$ and $Y$ are independent without using the uniqueness of the MGF

Let $(X, Y)$ be an absolutely continuous random vector and $M_{X, Y}$ be the bivariate moment generating function of $(X, Y)$. I want to show that $X$ and $Y$ are independent if and only if $$M_{X, Y}\...
Cyclotomic Manolo's user avatar
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How do we prove that $X\perp Y$ when $f_{X,Y}(x,y) = \frac{1}{2\pi}e^{\frac{-1}{2}(x^2+y^2)}$? [closed]

I was wondering if $$f_{X,Y}(x,y)=\frac{1}{2\pi}e^{\frac{-1}{2}(x^2+y^2)}$$ Is sufficient to show that X and Y have independent standard normal distributions? If not what else would I need to show? ...
edster101's user avatar
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CDF and PDF of dependent variables originating from a real life shock models

Suppose there are two sources of shocks and a system with two components. Shock A can affect the first component and Shock B can affect both the components. I am trying to find the joint distribution ...
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Under what conditions does a Normal random variable have approximately the same conditional distribution at every value of a discrete variable?

Let $X$ be a random variable with known distribution $N(\mu, \sigma)$. Let $Y$ be a count variable, i.e., it is a discrete random variable with known finite expectation and finite variance, which ...
virtuolie's user avatar
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Expectation and variance of bivariate skew normal distribution

I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
user2167741's user avatar
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Joint distribution of number of trials needed to get 1 and 2 heads

I have the following random variables $X = $ number of trials needed to obtain the first head $Y = $ number of trials needed to get two heads in repeated tosses of a fair coin. How I cand find the ...
daniel's user avatar
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Interpretation if integrate a bivariate distribution with respect to one component only

while solving an integral problem for bivariate distribution, I encounter with some problem when interpreting the results. Let say that I have a bivariate continuous function $f(t,x)$ where $t$ and $x$...
AHMAD FAIZ BIN MOHD AZHAR MSC2's user avatar
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Bound on Jensen's gap for bivariate functions

I have recently been reading on Jensen's gap for univariate functions. Specifically, here, page 6, Theorem 2.1, an upper bound on Jensen's gap is provided that is based on absolute central moments. I ...
Mohammad Hussein Yoosefian Noo's user avatar
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Using Orthogonal Decomposition of Bivariate Normally Distributed X,Y Reveals Different Result Than Calculating Variance Directly

This is the exercise description: 6.5.4. Suppose X and Y are standard normal variables. Find an expression for P(X + 2Y ≤ 3) in terms of the standard normal distribution function Φ, (a) in case X and ...
BurgerMan's user avatar
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Joint Distribution Function of a Joint Density Function (exponential)

I have a density function $f(x,y)= 2e^{-x-y}$ for $0<x<y<\infty$ and $0$ elsewhere. What is the joint distribution function? So far I have calculated $F(x,y)= \int_0^x \int_0^yf(x',y') dy'dx' ...
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Find joint distribution from transformation

Let $U_1$ and $U_2$ be uniformly distributed on $(0,1]$. Let $$ X_1 = \sqrt{-2\log(U_1)} \cos(2 \pi U_2), $$ and $$ X_2 = \sqrt{-2\log(U_1)} \sin(2 \pi U_2) $$ Find the joint distribution of $X_1$ and ...
sucksatmath's user avatar
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How do I find the joint pdf of bivariate normal distribution from scratch given linear correlation coefficient and their marginal distributions?

I'm taking a stat class and the professor just wrote down the joint pdf of a bivariate normal distribution, even though the pdf looks "intuitive" but I'm not able to derive the formula for ...
aroma's user avatar
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Clarification - DeGroot, Probability, Mixed Bivariate Distribution, what does this line mean?

I understand how the probability has been calculated in both orders here. But I dont understand the highlighted part. What does it mean that the sum is $0$? If it weren't $0$, would we then not be ...
thecountofmontecristo's user avatar
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if (x y) ~ bivariate normal (0, 0, 1, 1, ρ), show that q = (x^2 −2ρxy+y^2)/ 1−ρ^2 is distributed as chi square (2 degrees of freedom).

Here X and Y follow bivariate normal distribution and Q is a new variable including X and Y. Prove that the new variable Q follows chi square distribution with 2 degree of freedom.
Soham Raut's user avatar
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Rate of convergence involving bivariate normal

Suppose $Z_1$ and $Z_2$ are bivariate normal, marginally standardized, and has correlation $\rho$. Fix $z_0>0$. I am interested in finding an estimate of the rate of convergence the probability $P(...
Uchiha's user avatar
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Joint distribution of bivariate normal and bernoulli

Let (X,Y) be a bivariate normal and Z follow Bernoulli distribution and be independent of (X,y). Its mean $(X,Y)\sim N(\mu,\sigma I)$ and $Z \sim Ber(p)$. How can I find the joint distribution of them?...
Long Tuấn's user avatar
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Summing indices during matrix multiplication

Lets say I have the following matrix of probabilities that I see either or both a cat or a dog on any given day. The probability of seeing a cat or a dog are not independent but generated via some ...
Tommy Glizda's user avatar
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Estimating one of the means of a bivariate Gaussian when the two means are unknown

Suppose we want to estimate a single mean $\mu_1$ of a bivariate Gaussian, whose covariance matrix is known, but the means $\mu_1$ and $\mu_2$ are unknown. Let $N$ be the number of joint samples. If ...
Daniel S.'s user avatar
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Bivariate distribution of angles between two vectors and random vector

Let $Z_1 \in \mathbb{R}^d$ and $Z_2\in\mathbb{R}^d$ be two nonzero vectors. Let $X$ be a random vector distributed uniformly on the hypersphere $\mathbb{S}^{d-1}$. In the case $d=2$, the marginal ...
Student's user avatar
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Computing the probability using a joint density function

I am in the middle of an exercise dealing with two random variables $(X,Y)$ having the bivariate normal density function: $$f_{(X,Y)}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\exp {\left[-\frac{1}{2(1-\rho^...
Quasar's user avatar
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E[XY] where X and Y are the **sign functions** of standard normal distributions

The following question is from the book: "150 Most Frequently Asked Questions on Quant Interviews" By Stefanica, Radoicic, and Wang. Let $X$ and $Y$ be standard normal variables with joint ...
JoeIsh's user avatar
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Find the CDF of Z when Z=XY for a given joint PDF for X and Y

The joint probability density function of random variables $ X$ and $ Y$ is given by $$ f_{X,Y}(x, y) = 2(1-y) \quad \text{if} \quad 0 \leq x \leq 1, 0 \leq y \leq 1 $$ $$ f_{X,Y}(x, y) = 0 \quad \...
ripbozo's user avatar
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Conditional mean and variance of two Gaussians given one is larger.

Given two Gaussian random variables $Z:=(Z_1,Z_2)\sim N(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{2\times 2}$, is there an explicit expression for the mean and variance of $Z$ conditioned on $Z_1>Z_2$...
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Finding Correlation of a Bivariate Function

Here is a bivariate function with the variables $i,j$ is given as: $f(i,j) = Wij(i+j),$ $0<i<1,$ $0<j<1.$ Where $W$ is just a constant. Assume that $M=\min(I,J)$ and $N = \max(I,J).$ ...
Ab2020's user avatar
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If there's another variable that controls whether heads outcomes are allowed, what distribution would that coin toss follow?

For a usual coin with probability of heads $P(H)$, the distribution of the numbers of heads after $n$ tosses follows a binomial distribution. The coin can be biased or unbiased. But suppose there's ...
HuN's user avatar
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Expectation in restricted bivariate distribution

I'm trying to follow the solution of this exercise: link. It says: Let $X$ and $Y$ be two jointly continuous random variables with joint PDF $ \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{...
Oliver Mohr Bonometti's user avatar
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2 answers
105 views

Mean, Variance, and Conditional PMF from Poisson Process

In good years, quarrels between Tolstoy and his wife occurred according to a Poisson process with rate λ = 5 per month. In bad years, it was a Poisson process with rate μ = 10 per month. Suppose each ...
Pierre's user avatar
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1 answer
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How to derive conditional expectation E[X|Y=y] if X and Y follow bivariate normal distribution

I am struggling to derive $E[X|Y=y]=\mu_X+\sigma_X\rho(\frac{y-\mu_y}{\sigma_Y})$ when X and Y follow bivariate normal distribution. I have read this, but I don't get how to get the following steps: $$...
jasmine's user avatar
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Conditional distribution of a bivariate Gaussian/normal distribution

A bivariate Gaussian pdf is $$ f_{X, Y}(x, y)=\frac{1}{2 \pi|\Sigma|^{1 / 2}} \exp \left(-\frac{1}{2}\left[x-m_1, y-m_2\right] \Sigma^{-1}\left[x-m_1, y-m_2\right]^T\right) $$ where $\Sigma=\left[\...
Blahblahblacksheep's user avatar
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1 answer
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Why is the following variable a mixed random variable?

Let X∼N(0,1) and W∼Bernoulli(1/2) be independent random variables. Define the random variable Y as a function of X and W: Find the PDF of Y and X+Y. I don't understand why Z turned out to be a mixed ...
Sam's user avatar
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Bivariate normal CFD approximation using characteristic function

The normal distribution CFD can be approximated using $$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$ where the characteristic ...
Ruan's user avatar
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3 votes
2 answers
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The root of the sum of two normally distributed variables

Given $X,Y \sim \mathcal{N}(0,1)$ and $Z=\sqrt{X^2 + Y^2}$, find the PDF of $Z$. I know from digging around that this will follow a Rayleigh distribution since the sum of two squared normally ...
J N's user avatar
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0 answers
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Log supermodularity property of Bivariate Normal CDF

I am trying to understand for bivariate normal variables, when their correlation increases, can we show in certian sense the distribution is concentrated on the 45 degree line. For example, let $F(X,Y,...
user1067394's user avatar
1 vote
1 answer
68 views

Probability of 2 dimensional random variable.

Two balls are selected at random without replacement from a box that contains 3 blue, 2 red, and 3 green balls. If $X$ is the number of blue balls selected and $Y$ is the number of red balls selected. ...
Ganesh Kumar's user avatar
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1 answer
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finding constant in a bivariate join pdf

I'm given a $$f_{X,Y}(x,y) = \begin{cases} cx, & \text{x > 0, y > 0, 1}\ \leq \ x+y \ \leq 2, \\ 0, & \text{elsewhere.} \end{cases}$$ and trying to find a the constant $c$. I've set ...
Xenotion's user avatar
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Marginal density equal to zero everywhere.

I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
MarkH9664's user avatar
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1 answer
134 views

Show that U and V are independent.

Suppose we have (X, Y ) be a bivariate normal vector with mean vector µ = (0, 0) and covariance matrix $$ Σ= \begin{bmatrix}2 & 1 \\1 & 2 \\\end{bmatrix} $$ I'm given: $$ A^TΣA=\begin{bmatrix}...
murpw2011's user avatar
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The bivariate normal distribution and its ellipse

Let $f(x,y)$ be a bivariate normal p.d.f. and let $c$ be a positive constant so that $c < \left(2\pi \sigma_{X} \sigma_{Y}\sqrt{1-\rho^2}\right)^{-1}$. Show that $c=f(x,y)$ defines an ellipse in ...
Paul Ash's user avatar
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0 votes
2 answers
313 views

Using general bivariate gaussian to extract marginal PDF from given bivariate PDF

I had a homework question to find the marginal probability density functions, $p_X(x)$ and $p_Y(y)$, given a join probability density function $p_{XY}(x,y)$. I have solved the problem by integration i....
Aserian's user avatar
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0 answers
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Distribution of a bivariate Gaussian

Let $X,Y$ be a bivariate normal distribution with correlation $\rho$, means $\mu_1 \mu_2$ and variances $\sigma_1^2, \sigma_2^2$. Is there an expression for $$P(X \ge 0, Y \ge 0)$$ that only uses $\...
Claudio Moneo's user avatar
1 vote
1 answer
36 views

Confusing in limits of bivarite distribution and contradiction in related question

This question is from "Mathematical Statistics with application by Wackerly and Mendelhall" $8th$ edition page $233$ ,question $5.5$ : $f(y_1,y_2)=3y_1 , 0 \leq y_2 \leq y_1 \leq1$ and $0$ ,...
user avatar
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1 answer
55 views

confusion on determining integral limits in bivarite distribution

$f(x_1,x_2)= 8x_1x_2 , 0<x_1<x_2<1 $ $\text{and}$ $0 , \text{elsewhere}$ I want to find $E(X_1X_2^2).$ According to the book: $$E(X_1X_2^2) = \int_{0}^{1}\int_{0}^{x_2} (x_1x_2^28x_1x_2)...
Not a Salmon Fish's user avatar
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1 answer
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Pdf of sum of independent rvs is the convolution of pdfs proof

I am trying to prove the statement in the title. However, I want some help to make my derivation mathematically rigorous. I have $X_{1}$ and $X_{2}$ which are two independent rvs. In addition, I have $...
makala's user avatar
  • 135
1 vote
1 answer
207 views

Covariance for a bivariate normal distribution

I have a question concerning the bivariate normal distribution where I have been able to prove that the first covariance we are asked for is equal to $0$, but was not sure if this would be implying ...
Stiven G's user avatar
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0 answers
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Find $a$ such that the $P(X \in s) = 0.95$ where $ s = \{(x_1,x_2)^T \in \Bbb R : -a \leq x_1 \leq a \} $. Standard bivariate normal distribution.

I have a bivariate normal distribution, where $ X = (X_1, X_2)^T$. The means are given by $\mu = (0,0)^T$ and the covariance matrix is $ \sum = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} ...
user898975's user avatar
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1 answer
585 views

Prove Independence of a Bivariate Normal Distribution

Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent. I'm feeling pretty stumped by this question, ...
  skwirlyburd's user avatar
1 vote
0 answers
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Solution to bivariate normally distributed problem

Is anyone able to solve the following integral with a normal bivariate PDF, or at least to clarify if there exist a closed form solution? Thanks in advance. \begin{equation} \frac{1}{2\pi} \int_{-\...
Eastwood94's user avatar
2 votes
1 answer
573 views

Given probability density $f(x, y)=12xy(1-y)$, find the joint density of $U=XY^2$ and $V=Y$ and the marginals

Let $X$ and $Y$ be random variables with probability density $f(x, y)=12xy(1-y)$ if $0<x<1, 0<y<1$. Find the joint probability density of $U=XY^2$ and $V=Y$. Find the marginal density of $...
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